before deviding main file in many

images/covering.pdf_tex

@@ -58,13 +58,13 @@
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     \put(0,0){\includegraphics[width=\unitlength,page=3]{covering.pdf}}%
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   \end{picture}%
 \endgroup%

images/covering.svg

@@ -106,12 +106,12 @@
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-         id="flowPara743">\shortstack{$ t^{-1}\alpha_1, t^{-1}\alpha_2, \cdots, t^{-1}\alpha_n$}</flowPara></flowRoot>    <flowRoot
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images/knot_complement.pdf_tex

@@ -54,13 +54,13 @@
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images/knot_complement.svg

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lectures_on_knot_theory.tex

@@ -69,7 +69,8 @@
 
 \newcommand{\sdots}{\smash{\vdots}}  
 
-
+\DeclareRobustCommand\longtwoheadrightarrow
+     {\relbar\joinrel\twoheadrightarrow}
 
 
 
@@ -79,8 +80,11 @@
 
 \DeclareMathOperator{\Hom}{Hom}
 \DeclareMathOperator{\rank}{rank}
-\DeclareMathOperator{\Gl}{Gl}
+\DeclareMathOperator{\ord}{ord}
+\DeclareMathOperator{\Gl}{GL}
+\DeclareMathOperator{\Sl}{SL}
 \DeclareMathOperator{\Lk}{lk}
+\DeclareMathOperator{\pt}{\{pt\}}
 
 
 \titleformat{\section}{\normalfont \fontsize{12}{15} \bfseries}{%
@@ -562,7 +566,7 @@ If $K$ is a knot, then $n$ is necessarily even, and so $\Delta_K(t^{-1}) = t^{-n
 \begin{lemma}
 \begin{align*}
 \frac{1}{2} \deg \Delta_K(t) \leq g_3(K),
-\text{ where } deg (a_n t^n + \cdots + a_1 t^l )= k - l.
+\text{ where } deg (a_n t^n + \dots + a_1 t^l )= k - l.
 \end{align*}
 \end{lemma}
 \begin{proof}
@@ -622,12 +626,10 @@ Figure 8 knot is negative amphichiral.
 %
 \section{Concordance group \hfill\DTMdate{2019-03-18}}
 \begin{definition}
-A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
-A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
-\end{definition}
-\begin{definition}
 Two knots $K$ and $K^{\prime}$ are called (smoothly) concordant if there exists an annulus $A$ that is smoothly embedded in ${S^3 \times [0, 1]}$ such that 
-${\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}}$.
+\[
+\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}.
+\]
 \end{definition}
 
 \begin{figure}[h]
@@ -637,6 +639,10 @@ ${\partial A = K^{\prime} \times \{1\} \; \sqcup \; K \times \{0\}}$.
 \resizebox{0.8\textwidth}{!}{\input{images/concordance.pdf_tex}}
 }
 \end{figure}
+\begin{definition}
+A knot $K$ is called (smoothly) slice if $K$ is smoothly concordant to an unknot. \\
+A knot $K$ is smoothly slice if and only if $K$ bounds a smoothly embedded disk in $B^4$.
+\end{definition}
 
 
 
@@ -681,9 +687,9 @@ Remark: $K \sim K^{\prime} \Leftrightarrow K \# -K^{\prime}$ is slice.
 \\
 \\
 \noindent
-Let $\Omega$ be an oriented \\
+Let $\Omega$ be an oriented four-manifold. \\
 ???????\\
-Suppose $\Sigma$ is a Seifert matrix with an intersection form ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z}$ (i.e. there are cycles). \\
+Suppose $\Sigma$ is a Seifert surface and $V$ a Seifert form defined on $\Sigma$: ${(\alpha, \beta) \mapsto \Lk(\alpha, \beta^+)}$. Suppose $\alpha, \beta \in H_1(\Sigma, \mathbb{Z})$ (i.e. there are cycles). \\
 ??????????????\\
 $\alpha, \beta \in \ker (H_1(\Sigma, \mathbb{Z}) \longrightarrow H_1(\Omega, \mathbb{Z}))$. Then there are two cycles $A, B \in \Omega$ such that $\partial A = \alpha$ and $\partial B = \beta$. 
 Let $B^+$ be a push off of $B$ in the positive normal direction such that 
@@ -692,6 +698,7 @@ Then
 $\Lk(\alpha, \beta^+) = A \cdot B^+$
 %
 %
+%ball_4_alpha_beta.pdf
 \\
 \section{\hfill\DTMdate{2019-04-08}}
 %
@@ -727,7 +734,7 @@ of $H_2(Y, \mathbb{Z}$, then $A = (b_i, b_y)$ \\??\\ is a matrix of intersection
 \begin{align*}
 \quot{\mathbb{Z}^n}{A\mathbb{Z}^n} \cong H_1(Y, \mathbb{Z}).
 \end{align*}
-In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z}$.\\
+In particular $\mid \det A\mid = \# H_1(Y, \mathbb{Z})$.\\
 That means - what is happening on boundary is a measure of degeneracy.  
 
 \begin{center}
@@ -750,16 +757,17 @@ H_1(Y, \mathbb{Z}) &
 \end{tikzcd}
 $(a, b) \mapsto aA^{-1}b^T$
 \end{center}
-
+?????????????????????????????????\\
+\noindent
 The intersection form on a four-manifold determines the linking on the boundary. \\
 
 \noindent
 Let $K \in S^1$ be a knot, $\Sigma(K)$ its double branched cover. If $V$ is a Seifert matrix for $K$, then 
 $H_1(\Sigma(K), \mathbb{Z}) \cong  \quot{\mathbb{Z}^n}{A\mathbb{Z}}$ where 
-$A = V \times V^T$, where $n = \rank V$.
+$A = V \times V^T$, $n = \rank V$.
 %\input{ink_diag}
 \begin{figure}[h]
-\fontsize{40}{10}\selectfont
+\fontsize{20}{10}\selectfont
 \centering{
 \def\svgwidth{\linewidth}
 \resizebox{0.5\textwidth}{!}{\input{images/ball_4.pdf_tex}}
@@ -777,20 +785,33 @@ Let $X$ be the four-manifold obtained via the double branched cover of $B^4$ bra
 \item The intersection form on $X$ is $V + V^T$.
 \end{itemize}
 \end{fact}
+\begin{figure}[h]
+\fontsize{20}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.5\textwidth}{!}{\input{images/ball_4_pushed_cycle.pdf_tex}}
+\caption{Cycle pushed in 4-ball.}
+\label{fig:pushCycle}
+}
+\end{figure}
 \noindent
 Let  $Y = \Sigma(K)$. Then:
-\begin{flalign*}
+\begin{align*}
-H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) \longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}&
+H_1(Y, \mathbb{Z}) \times H_1(Y, \mathbb{Z}) &\longrightarrow \quot{\mathbb{Q}}{\mathbb{Z}}
 \\
-(a,b) \mapsto a A^{-1} b^{T},\qquad
+(a,b) &\mapsto a A^{-1} b^{T},\qquad
-A = V + V^T&
+A = V + V^T.
-\\
+\end{align*}
-H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}&\\
+????????????????????????????
-A \longrightarrow BAC^T \quad \text{Smith normal form}&
+\begin{align*}
-\end{flalign*}
+H_1(Y, \mathbb{Z}) \cong \quot{\mathbb{Z}^n}{A\mathbb{Z}}\\
+A \longrightarrow BAC^T \quad \text{Smith normal form}
+\end{align*}
 ???????????????????????\\
 In general 
 
+%no lecture at 29.04
+
 \section{\hfill\DTMdate{2019-05-20}}
 
 Let $M$ be compact, oriented, connected four-dimensional manifold. If ${H_1(M, \mathbb{Z}) = 0}$ then there exists a 
@@ -1040,6 +1061,7 @@ $2 \sum\limits_{k_i \text{ odd}} \epsilon_i$. The peak of the signature function
 \end{theorem}
 \end{proof}
 \section{\hfill\DTMdate{2019-05-27}}
+
 ....
 \begin{definition}
 A square hermitian matrix $A$ of size $n$.  
@@ -1063,8 +1085,12 @@ Remove from $\Delta$ the two self intersecting and glue the Seifert surface for
 \begin{example}
 The knot $8_{20}$ is slice: $\sigma \equiv 0$ almost everywhere but $\sigma(e^{\frac{ 2\pi i}{6}}) = + 1$.
 \end{example}
-\subsection{Surgery}
+%ref Structure in the classical knot concordance group
-Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \{pt\}]}$ and ${\beta=[\{pt\} \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. 
+%Tim D. Cochran, Kent E. Orr, Peter Teichner
+%Journal-ref: Comment. Math. Helv. 79 (2004) 105-123
+\subsection*{Surgery}
+%Rolfsen, geometric group theory, Diffeomorpphism of a torus, Mapping class group
+Recall that $H_1(S^1 \times S^1, \mathbb{Z}) = \mathbb{Z}^3$. As generators for $H_1$ we can set ${\alpha = [S^1 \times \pt]}$ and ${\beta=[\pt \times S^1]}$. Suppose ${\phi: S^1 \times S^1 \longrightarrow S^1 \times S^1}$ is a diffeomorphism. 
 Consider an induced map on homology group: 
 \begin{align*}
 H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad p, q \in \mathbb{Z},\\
@@ -1075,9 +1101,30 @@ H_1(S^1 \times S^1, \mathbb{Z}) \ni \phi_* (\alpha) &= p\alpha + q \beta, \quad
      r & s
   \end{pmatrix}
 \end{align*} 
-
+As $\phi_*$ is diffeomorphis, it must be invertible over $\mathbb{Z}$. Then for a direction preserving diffeomorphism we have  $\det \phi_* = 1$. Therefore $\phi_* \in \Sl(2, \mathbb{Z})$.
 \end{theorem}
 
+\vspace{10cm}
+\begin{theorem}
+Every such a matrix can be realized as a torus. 
+\end{theorem}
+\begin{proof}
+\begin{enumerate}[label={(\Roman*)}]
+\item
+Geometric reason
+\begin{align*}
+\phi_t: 
+S^1 \times S^1 &\longrightarrow S^1 \times S^1 \\
+S^1 \times \pt  &\longrightarrow \pt \times S^1 \\
+\pt  \times S^1 &\longrightarrow S^1 \times \pt \\
+(x, y) & \mapsto (-y, x)
+\end{align*}
+\item
+\end{enumerate}
+\end{proof}
+
+
+
 
 \section{balagan}
 
@@ -1128,4 +1175,87 @@ has a subspace of dimension $g_{\Sigma}$ on which it is zero:
      * & \dots & * & * & \dots & * 
  \end{pmatrix}_{2g_{\Sigma} \times 2g_{\Sigma}}
 \end{align*}
+
+
+\section{\hfill\DTMdate{2019-05-06}}
+
+\begin{definition}
+Let $X$ be a knot complement. 
+Then $H_1(X, \mathbb{Z}) \cong \mathbb{Z}$ and there exists an epimorphism 
+$\pi_1(X) \overset{\phi}\twoheadrightarrow \mathbb{Z}$.\\
+The infinite cyclic cover of a knot complement  $X$ is the cover associated with the epimorphism $\phi$. 
+\[
+\widetilde{X} \longtwoheadrightarrow X
+\] 
+\end{definition}
+%Rolfsen, bachalor thesis of Kamila
+\begin{figure}[h]
+\fontsize{10}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{1\textwidth}{!}{\input{images/covering.pdf_tex}}
+\caption{Infinite cyclic cover of a knot complement.}
+\label{fig:covering}
+}
+\end{figure}
+\begin{figure}[h]
+\fontsize{10}{10}\selectfont
+\centering{
+\def\svgwidth{\linewidth}
+\resizebox{0.8\textwidth}{!}{\input{images/knot_complement.pdf_tex}}
+\caption{A knot complement.}
+\label{fig:complement}
+}
+\end{figure}
+\noindent
+Formal sums $\sum \phi_i(t) a_i + \sum \phi_j(t)\alpha_j$ \\
+finitely generated as a $\mathbb{Z}[t, t^{-1}]$ module. 
+\\
+Let $v_{ij} = \Lk(a_i, a_j^+)$. Then
+$V = \{ v_ij\}_{i, j = 1}^n$ is the Seifert matrix associated to the surface $\Sigma$ and the basis $a_1, \dots, a_n$. Therefore $a_k^+ = \sum_{j} v_{jk} \alpha_j$. Then
+$\Lk(a_i, a_k^+)= \Lk(a_k^+, a_i) = \sum_j v_{jk} \Lk(\alpha_j, a_i) = v_{ik}$. 
+We also notice that $\Lk(a_i, a_j^-) = \Lk(a_i^+, a_j)= v_{ij}$ and 
+$a_j^- = \sum_k v_{kj} t^{-1} \alpha_j$.
+\\
+\noindent
+The homology of $\widetilde{X}$ is generated by $a_1, \dots, a_n$ and relations. 
+
+\begin{definition}
+The $\mathbb{Z}[t, t^{-1}]$ module $H_1(\widetilde{X})$ is called the Alexander module of knot $K$.
+\end{definition}
+\noindent
+Let $R$ be a PID, $M$ a finitely generated $R$ module. Let us consider
+\[
+R^k \overset{A} \longrightarrow R^n \longtwoheadrightarrow M,
+\] 
+where $A$ is a $k \times n$ matrix, assume $k\ge n$. The order of $M$ is the $\gcd$ of all determinants of the $n \times n$ minors of $A$. If $k = n$ then $\ord M = \det A$. 
+\begin{theorem}
+Order of $M$ doesn't depend on $A$. 
+\end{theorem} 
+\noindent
+For knots the order of the Alexander module is the Alexander polynomial.
+\begin{theorem}
+\[
+\forall x \in M: (\ord M) x = 0.
+\]
+\end{theorem}
+\noindent
+$M$ is well defined up to a unit in $R$.
+\subsection*{Blanchfield pairing}
+\section{balagan}
+\begin{theorem}
+Let $H_p$ be a $p$ - torsion part of $H$. There exists an orthogonal decomposition of $H_p$:
+\[
+H_p = H_{p, 1} \oplus  \dots \oplus H_{p, r_p}.
+\]
+$H_{p, i}$ is a cyclic module:
+\[
+H_{p, i} = \quot{\mathbb{Z}[t, t^{-1}]}{p^{k_i} \mathbb{Z} [t, t^{-1}]}
+\]
+\end{theorem}
+\noindent
+The proof is the same as over $\mathbb{Z}$.
+\noindent
+
 \end{document}
+